# Tagged Questions

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### Calculus in an abstract space

This is page 302 of PDE Evans, 2nd edition. DEFINITIONS. $\text{(i)}$ The Sobolev space $$W^{1,p}(0,T;X)$$ consists of all functions $\textbf{u} \in L^p(0,T;X)$ such that $\textbf{u}'$ exists in ...
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### Canonical Separation of variables

Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.
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### How to esimate $\inf\int|\nabla g|^p\,dx$

It is rather easy question but I'm already struggling with this problem for a long time. I'm trying to estimate the value $$\inf\int|\nabla g|^p\,dx$$ where $\mathbf{inf}$ is taken over all ...
### Integrating $\ln(1+|\ln|x||)$ in $B_1(0)$
I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$ Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and ...
You have $f \in C^\infty([0,1])$ with $f > 0$. Then $\sqrt{f}$ is easily seen to be differentiable . Prove that there exists a constant $C$ independent of $f$ such that: ...